Neutron Scattering

More documents

Recommendations

Info

y r f R(Q)- (du /dS2) ++ - u a b z1-4 7r ° MLQ) (d6/dS2) 1+7 r° M + . . . 2u bQ pB b M Note that the polarisation of the scattered beam is known a priori (only non-spin flip processes can occur), so that the experiment is being donc with a polarised beam, but without polarisation analysis . The flipping ration (16 .11) depends linearly on the magnetic structure factor, instead of quadratic as the scattered intensity . Therefore much smaller values of the magnetic structure factor can be determined. If the nuclear structure factor is known (e.g . from a prior neutron diffraction experiment), these measurements of the flipping ratio give access to a highly precise determination of the phase and magnitude of the magnetic structure factor . An example is given by the measurement of the form factor of chromium. Cr is the archetypal ifnerant antiferromagnet. Therefore the magnetisation density is very delocalised . As a consequence, the magnetic form factor drops extremely rapidly with increasing momentum transfer. In a recent synchrotron x-ray experiment, we could demonstrate that this form factor is spin only [11] . However, in a polarised neutron diffraction experiment we could show [12], that a magnetisation induced in the paramagnetic state by an extemal magnetic field is much more localised around the individual atoms . Therefore, the field-induced form factor decreases much slower, compare figure 16 .6 . It has a large contribution (60 %) of orbital angular momentum, quite in contrast to the form factor in the ordered state . By means of a Fourier transform or with the so-called Maximum Entropy Method a magnetisation density distribution within the unit cell can be reconstructed (compare figure 16 .7) . Such data are of utmost importance to test and improve modern band theories, such as the fully relativistic density functional theory and thus to obtain a better understanding of the metallic state .
0 .8 ' D3 ° Stassis 3d spin - - 3d orbital - 40% spin +60% orbital w = 0 .4 m 0 .0 0.0 0 .2 0.4 0.6 0 .8 1 .0 1/2d [Ä - '] Fie. 16.6 :Lnduced magneticformfactor ofCr for afield of4.6 T. Open andfilled circles are experimental values, the Lines are calculations for spin-, Orbital- and total moment. Fie. 16 .7: Projection ofthe induced magnetisation density distribution onto the plane. above : Fourier transform Below : Maximum entropy reconstruction 16 .4 Magnetie Phase Transitions Phase transitions can occur between different magnetic phases as a fonction of various thermodynamic parameters, such as magnetic field, temperature or pressure . Here we will restrict ourselves to the most simple case of a transition from a low temperature ferromagnetic (FM) or antiferromagnetic (AF) phase to a high temperature paramagnetic (PM) phase . First, we will discuss this phenomenon qualitatively, then introduce the quantitative description and fmally show just one example of a neutron diffraction study . The magnetic long range order discussed in section 16 .2 can only be stable, as long as the thermal energy kBT is small enough compared to the exchange interactions giving rise to magnetic order . At sufficiently high temperatures, entropy wins and the magnetic moments 16- 12
Page 1 and 2:
Forschungszentrum Jülich in der He
Page 4 and 5:
Forschungszentrum Jülich GmbH Inst
Page 6:
Contents . . 1 Neutron Sources H .
Page 10 and 11:
1 . Neutron Sources Harald Conrad 1
Page 12 and 13:
d) f, = v a, , q - ti = P a, - 0 .3
Page 14 and 15:
Period Example Flux (D [1013 1950-6
Page 16 and 17:
In stage 1 the primary proton knock
Page 18 and 19:
get as heat, the rest is transporte
Page 20 and 21:
down power and stronger absorption
Page 22 and 23:
Appendix Neutron Sources - an overv
Page 24:
led remotely. It is rauch more conv
Page 28 and 29:
2 . Properties of the neutron, elem
Page 30 and 31:
In the 60's the ferst high flux rea
Page 32 and 33:
Hot neutrons in reactors are obtain
Page 34 and 35:
elated values for the FRJ-2 reactor
Page 36 and 37:
2 .5 Scattering amplitude and cross
Page 38 and 39:
(A+k2 ) yr = u(Z:) yr V( u~r)= z "
Page 40 and 41:
_du _ mn \2 A2-~2z r2~ I(k' IVI k)I
Page 42 and 43:
E ô ô x z w 0 z _w U N Figure 2 .
Page 44 and 45:
scattering is observed with the ave
Page 46 and 47:
For a spherical electron distributi
Page 48:
A more thorough introduction to neu
Page 52 and 53:
3 . Elastie Seattering from Many-Bo
Page 54 and 55:
Q = Q = k2 + k2 - 2kk' cos2B (3 .3)
Page 56 and 57:
3 .2 Fundamental Scattering Theory
Page 58 and 59:
The meaning of (3 .l4) is immediate
Page 60 and 61:
1 . e. in the second approximation,
Page 62 and 63:
The saure problem will bc dealt wit
Page 64 and 65:
ALQ) = Y-bfe'Q'-rB(r-(n .a+m .b+p»
Page 66 and 67:
du LQ) = ALQJ2 = e'Q .A' . * -i e-
Page 68 and 69:
Tab . 3.1 : The scattering lengths
Page 70 and 71:
We note immediately that we should
Page 72 and 73:
These magnetic structures can be un
Page 74 and 75:
M1LQ)=Q-MQ)xQ M(Q)= IM (r)e i Q " d
Page 76 and 77:
Orbital magnetic spin angular ,,- m
Page 78 and 79:
aQ . r 3 àQ . N . QR . aQ .t k _ML
Page 80:
Schweika Analysis Polarization W .
Page 83 and 84:
ducing a complex component and were
Page 85 and 86:
ir v s ftft 3956 2 s 2 0.81[X] 80 c
Page 87 and 88:
Some polarizers using Bragg reflect
Page 89 and 90:
Different to the coherent nuclear s
Page 91 and 92:
where QBG denotes the background .
Page 93 and 94:
4 .4 Applications We now consider e
Page 95 and 96:
Fig. 4 .3 .4 probably represents th
Page 97 and 98:
. occur only in the non-spin flip c
Page 99 and 100:
The results for the magnetization d
Page 101 and 102:
a large solid angle with polarizers
Page 103 and 104:
scattering to spin-flip scattering
Page 105 and 106:
An experimental verification of thi
Page 108:
Correlation Functions Measured by S
Page 111 and 112:
In real liquids however usually an
Page 113 and 114:
The term in big parentheses of equa
Page 115 and 116:
4 0 -5 0 5 10 15 20 ho) [meV] Figur
Page 117 and 118:
Figure 5 .5 : The pair correlation
Page 119 and 120:
Figure 5 .6 : Partial differential
Page 121 and 122:
Then equation (5 .21) can be conver
Page 123 and 124:
The route from expression (5 .27) t
Page 125 and 126:
In addition it is often useful to d
Page 127 and 128:
2 . The pair correlation function h
Page 129 and 130:
G(r, t) G(r, t) i(Q, t) r Q i(Q, t)
Page 131 and 132:
Insertion of this result into (5 .6
Page 133 and 134:
With this example we can also demon
Page 136 and 137:
6. Continuum description : Grazing
Page 138 and 139:
Starting point is the exact atomic
Page 140 and 141:
For specular reflectivity measureme
Page 142 and 143:
v v z l 2r V(z) = 2 -2erfC~ with er
Page 144 and 145:
F{dV(z)/dz ) to calculate the refle
Page 146 and 147:
Equation (6 .13) also shows that th
Page 148 and 149:
ut is completely reflected . The cr
Page 150 and 151:
0.4° . The full dynamical theory c
Page 152 and 153:
1) If no magnetic induction B (whic
Page 154:
Diffractometer Gernot Heger
Page 157 and 158:
select a special wavelengths band (
Page 159 and 160:
" The direction of the reciprocal l
Page 161 and 162:
Bragg-intensities of single crystal
Page 163 and 164:
monochromator, on the mosaic spread
Page 165 and 166:
complex sample environments, such a
Page 167 and 168:
the diffraction plane) two further
Page 170:
Small-angle Scattering and Reflecto
Page 173 and 174:
L= (1) ( k ) a e -(k/k T ) 2 Ak (8
Page 175 and 176:
possible neutron wave lengths betwe
Page 177 and 178:
figure and with neutrons of about 1
Page 179 and 180:
1 .0 Danvinkurve 0 .8 0 .6 0 .4 0 .
Page 181 and 182:
tails. This reflection curve leads
Page 183 and 184:
For the Nickel film one gets a thic
Page 185 and 186:
I The rotor of a velocity selector
Page 187 and 188:
multiplier are arranged in a quadra
Page 190 and 191:
9 . Crystal spectrometer : triple-a
Page 192 and 193:
machine which will be followed by a
Page 194 and 195:
10 .3 Beaur shaping Due to the fact
Page 196 and 197:
G.k=zGz . (11) This case is fulfill
Page 198 and 199:
tribution of bisecting planes (ntos
Page 200 and 201:
directions which is exploited for t
Page 202 and 203:
Fig . 14) Dispersion surfaces for B
Page 204 and 205:
whereby E and v,, denote energy and
Page 206:
[1] B . N. Brockhouse, in Inelastic
Page 210 and 211:
10 . Time-of-flight spectrometers M
Page 212 and 213:
Tnloderator /K V7 2/m/s A/nm Aiw/eV
Page 214 and 215:
10.2 .1 Interpretation of spectra A
Page 216 and 217:
----~ Probe ' (Chopper 2) Zeit I Ch
Page 218 and 219:
the similar intensity distribution
Page 220 and 221:
is performed without Jacobian due t
Page 222 and 223:
10.2 .6 Crystal monochromators The
Page 224 and 225:
larger energy transfers . However i
Page 226 and 227:
10.3 Inverted TOF-spectrometer In t
Page 228 and 229:
constant offset to the TOF . To be
Page 230 and 231:
sample size of cm and a detection p
Page 232:
Contents 10 .1 Introduction . . . .
Page 236 and 237:
11 . Neutron spin-echo spectrometer
Page 238 and 239:
2 0 L Tr -0 L 2 ArDet Figure 11 .1
Page 240 and 241:
initial velocity-reassemble at the
Page 242 and 243:
distribution (current sheets) that
Page 244 and 245:
esults, where il is an irrelevant c
Page 246 and 247:
10000 - 8000 - co 6000 - C ô 4000-
Page 248 and 249:
e0 oX n .5 1 .0 1 .5 so 0 .0 0 .0 1
Page 250 and 251:
nel" coils) the required homogeneit
Page 252 and 253:
Figure 11 .9 : S(Q, t)IS(Q) of a 2
Page 254:
Contents 11 .1 Introduction . . . .
Page 258 and 259:
12 . Structure determination G . He
Page 260 and 261:
Important: Thc measured Bragg inten
Page 262 and 263:
determination of accurate atomic pa
Page 264 and 265:
The knowledge of the overall occupa
Page 266 and 267:
(even anharmonic) effects. This com
Page 268 and 269:
Fig . 4 . Coordination ofthe D20 mo
Page 270 and 271:
fenoelectric phase of orthorhombic
Page 272 and 273:
(c) and (d) calculated densities at
Page 274:
13 Inelastic Neutron Scattering Pho
Page 277 and 278:
Figurel Schematical drawing of a ge
Page 279 and 280:
equilibrium position the atom is in
Page 281 and 282:
s - ' f N .mwn .I )oo K L .~um..un
Page 283 and 284:
esults from the quantum mechanics o
Page 285 and 286:
y absorption . 13 .1 .3 Magnetic in
Page 287 and 288:
Figure 7 The plane in reciprocal sp
Page 289 and 290:
5 100s1 [OSl ] IOSSI [SSS y i 4 0 r
Page 291 and 292:
dynamics of ionic compounds with th
Page 293 and 294:
16 14 12 F.. . 10 ô 8 q 6 cr 4 G.
Page 295 and 296:
21 [ooç 1 " - 19 18 17 16 "" 298 K
Page 297 and 298:
n = v - cap[i(pga . - wt)] (13 .22)
Page 299 and 300:
Y c 3 c Cc E Figure 16 Magnon dispe
Page 301 and 302:
. . 2 H . Bilz and W . Kress, Phono
Page 304 and 305:
14 . Soft Matter : Structure Dietma
Page 306 and 307:
length density of a sample against
Page 308 and 309:
2 P(Q) = 1 ~exp(-Ii - j'6 Q2 ) (14
Page 310 and 311:
fraction. In the region of large Q
Page 312 and 313:
with the partial structure factor S
Page 314 and 315:
and which lead to the scattering cr
Page 316 and 317: For the intramolecular and intermol
Page 318 and 319: 500 450 -400 U 0 ;, 350 42, 300 s~.
Page 320 and 321: scaling behavior of the susceptibil
Page 322: 15 Polymer Dynamics Dieter Richter
Page 325 and 326: constraints due to the mutual inter
Page 327 and 328: (15 .3) T ß ( E)=T ô exp E k,T -
Page 329 and 330: This incoherent approximation does
Page 331 and 332: 10 an 0 2 0 Figure 15.4 : Relaxatio
Page 333 and 334: In order to test Eq .[15 .9] the re
Page 335 and 336: Arrhenius temperature dependence wi
Page 337 and 338: 1 3kBT ~ z 2 72 2 p2 a = f2 N2 P =W
Page 339 and 340: S (Q,t) = 12 fdu exp ~ -u_ (S2Rt) v
Page 341 and 342: We now use these data, in order to
Page 343 and 344: c 0 .8 0 .6 00 .055 x 0 .1 110 .14
Page 345 and 346: small but systematic deviations fro
Page 347 and 348: S(Q,t) is predicted . Such a platea
Page 349 and 350: only one single parameter, the tube
Page 351 and 352: confinement of the relaxation of la
Page 354: 16 Magnetism Thomas Brückel
Page 357 and 358: the ground state of metallic magnet
Page 359 and 360: c~ ~ o 0 `,~i) g',, b o 0 Q! b o 0
Page 361 and 362: only "sec" this component and not t
Page 363 and 364: 200 ¬ 400 500 600 700 E 3 QI J (h0
Page 365: Fie. 16.4: Scattering geometry for
Page 369 and 370: Here, J;j denotes the exchange cons
Page 371 and 372: Fig. 16.8 : Contour plot of the mag
Page 373 and 374: 0.0001 0.001 0 .01 0.1 2 Fig. 16 .1
Page 376: 17 Translation and Rotation M . Pra
Page 379 and 380: 17.1 Introduction Atomic and molecu
Page 381 and 382: 17 .2.2 Diffusion, microscopic appr
Page 383 and 384: 0.6 H20 2 x2tÄ z ) Figure 17 .2 :
Page 385 and 386: The scattering function of a polycr
Page 387 and 388: e (2) 1 (j 1, 0 > - 0,1 >), respect
Page 389 and 390: leads to the eigenvalue problem ~q
Page 391 and 392: allow a good determination of the E
Page 393 and 394: m Eu W Figure 17 .7 : Eigenenergies
Page 395 and 396: -so o so Energy transfer (NeV) Ener
Page 397 and 398: Bibliography [1] T. Springer, Quasi
Page 400 and 401: 18 . Texture in Materials and Earth
Page 402 and 403: Further important structure related
Page 404 and 405: The axes of the cartesian Kp coordi
Page 406 and 407: 4.0 Experimental Texture Analysis T
Page 408 and 409: Fig. 18.14: Variation of reflection
Page 410 and 411: measurements can be performed in tr
Page 412 and 413: Apart from the global description o
Page 414 and 415: In the following, two experimental
Page 416 and 417:
The neutron diffraction pole figure
Page 418 and 419:
dependent pole densities, and (4) g
Page 420 and 421:
Schriften des Forschungszentrums J
Page 423:
Forschungszentrum Jülich in der He
show all

Neutron Scattering

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?